Problem 38 - Fundamentals of Mathematics

Question

Add the numbers. $$ \begin{array}{l} 30,188 \\ 79,731 \\ 16,600 \\ 66,085 \\ 39,169 \\ 95,170 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The sum is 326,943.

Step 1: Align the Numbers for Addition

Start by writing each number in a column, ensuring that each digit is aligned by its place value (thousands, hundreds, tens, units). This way, it's easier to keep track of carrying over numbers if needed.

Step 2: Add the Units Column

Add the digits in the units column (right-most column). The numbers are 8, 1, 0, 5, 9, and 0. The sum is \(8 + 1 + 0 + 5 + 9 + 0 = 23\). Write down the 3 in the units place and carry over the 2 to the tens column.

Step 3: Add the Tens Column

Now, add the digits in the tens column along with any carry-over. The numbers are 8, 3, 0, 8, 6, and 7, and carry-over is 2 from the previous step. The sum is \(8 + 3 + 0 + 8 + 6 + 7 + 2 = 34\). Write down the 4 in the tens place and carry over the 3 to the hundreds column.

Step 4: Add the Hundreds Column

Add the digits in the hundreds column with the carry-over. The numbers are 1, 7, 6, 0, 1, 1, and carry-over 3. The sum is \(1 + 7 + 6 + 0 + 1 + 1 + 3 = 19\). Write down the 9 in the hundreds place and carry over the 1 to the thousands column.

Step 5: Add the Thousands Column

Add the digits in the thousands column, including any carry-over. The numbers are 0, 9, 6, 6, 9, 5 and carry-over 1. The sum is \(0 + 9 + 6 + 6 + 9 + 5 + 1 = 36\). Write down the 6 in the thousands place and carry over the 3 to the ten thousands column.

Step 6: Add the Ten Thousands Column

Finally, add the digits in the ten thousands column with the carry-over. The numbers are 3, 7, 1, 6, 3, 9 and carry-over 3. The sum is \(3 + 7 + 1 + 6 + 3 + 9 + 3 = 32\). Since this is the last column, write down the complete number 32 as the starting of the result.

Key Concepts

Column Addition

Column addition is a reliable method to add large numbers together by organizing and aligning them for easier computation. To use column addition:

• Write each number one beneath the other, aligning the digits according to their place values (units, tens, hundreds, etc.).
• Ensure the numbers are straightly aligned so that each column represents a different place value.
• Sum the numbers starting from the rightmost column, moving towards the left.

You may notice some similarities to how you solve equations. The main advantage of column addition is maintaining the values in a structured format, making it easier to keep track of your sums. This method is particularly useful when dealing with long lists of numbers and helps to visualize the addition of each digit precisely.

Place Value

Place value represents the value of a digit based on its position within a number. Understanding place value is crucial in addition to correctly align digits:

• Digits on the right have the lowest place value, starting from the units, then tens, hundreds, and so on.
• A digit's value is its base value multiplied by the power of its place value. For example, in the number 2300, the '2' is in the thousand's place, so its value is 2,000.

Correctly aligning digits by place value ensures that units are added to units, tens to tens, and so on. This prevents calculation errors and ensures a smooth transition through each step of the addition process.

Carrying Over

Carrying over is a crucial step when sums of digits in a column exceed 9. This often occurs in multi-digit addition:

• When a column's sum is 10 or more, write down only the second digit (the remainder) in the current place value of your answer.
• The first digit (the carry-over) moves to the left, added to the next column's sum.
• This process ensures accuracy across all columns of the numbers being added.

Carrying over helps to manage larger sums effectively, distributing excess amounts correctly to higher place values. This step is repeated in each column as needed to maintain correct place value across the entire addition.

Units Column

The units column is the first column on the right side when performing column addition, representing the smallest place value:

• The units column consists of the numbers that have a place value of one (1).
• Begin the addition process from the units column and work leftwards incrementally.
• Any carry-over from the units column affects the tens column directly to the left.

The units column is the starting point of your addition process. Managing it properly sets a solid foundation for the rest of the columns. Since every column's addition depends on how well the previous column's sums are calculated, starting strong here is essential.